THE AXIOM OF CHOICE FOR FINITE SETS
R. L. BLAIR AND M. L. TOMBER
If ï is a family of nonempty sets, then by a choice function on S
we mean a function /on í such that f(S)ÇzS for every SÇzS. The
axiom of choice for finite sets may be formulated as follows:
(ACF) If S is any family of nonempty finite sets, then there exists a
choice function on S.
We shall denote the axiom of choice (for families of arbitrary nonempty sets) by (AC). It is easy to see that the ordering principle
(which asserts that every set can be totally ordered) implies (ACF).1
On the other hand, Mostowski
[4] has shown that (relative to a
suitable system of axiomatic set theory) the ordering principle is
actually weaker than (AC) so that (ACF), while it is surely a consequence of (AC), is not equivalent to (AC) [4, Korollar II, p. 250].
The object of the present note is to obtain equivalent formulations
of (ACF) of the Zorn's lemma type. In fact, in §1 we show that
(ACF) is equivalent to both (ZLF1) and (ZLF2) below; the latter
are obtained by restricting two familiar "maximal element" forms of
Zorn's lemma to a special class of partially ordered sets, namely, to
those "with finitary covers" (see the definition below). It is noteworthy, however, that not every form of Zorn's lemma, when so restricted, is equivalent to (ACF). For example, in §2 we observe that
if the statement
of either (ZLF1) or (ZLF2) is modified merely by
replacing the hypothesis of a least upper bound by that of an upper
bound, then the resulting formulation
is equivalent,
not to (ACF),
but to (AC) itself. And furthermore,
when one formulates the natural
"maximal chain" analogue of (ZLF1) and (ZLF2), one finds that the
result is again equivalent not to (ACF) but to the full axiom of choice
(AC).
The first named author is pleased to record his indebtedness
to
Professor Herman Rubin for a number of instructive
conversations
on the subject of this note.
1. Zorn's lemma formulations of (ACF). Let P be a partially
ordered set. For x, y(EP, define x~y in case either (i) x = y or (ii) x
Received by the editors May 22, 1959.
1 Tarski [9, p. 82] attributes this observation to Kuratowski;
for certain related
remarks, see [2; 3], and [lO]. For a study of certain variants of (ACF) (in which
cardinality restrictions are placed on the sets in S), see Mostowski [S], Szmielew
[8], and Sierpinski [7].
222
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THE AXIOM OF CHOICE FOR FINITE SETS
223
and y have no common upper bound. It is clear that the relation ~ is
both reflexive and symmetric.
If x, yEP, then x covers y in case x>y and x>z>y
for no zEPIf xEP, then we shall denote by C(x) the set of all elements of P
that cover x.
It will be convenient
to introduce the following definition:
Definition.
We say that P is a partially ordered set with finitary
covers in case for each nonmaximal element x£P the following condi-
tions hold:
(Fl) C(x) is not empty.
(F2) If aEC(x), then there exists a positive integer n(a) such that
any sequence a = ai-~a2,~as~
• • • in C(x) contains at most n(a) distinct terms.
(F3) If X is a nonempty subset of C(x) and if every pair of elements of X have a common upper bound, then X has a least upper
bound.
We can now formulate
the following
restricted
forms
of Zorn's
lemma:
(ZLF1) If P is a partially ordered set with
every well-ordered subset of P has a least upper
a maximal element.
(ZLF2) If P is a partially ordered set with
every totally ordered subset of P has a least upper
a maximal element.
Theorem
finitary covers, and if
bound, then P contains
finitary covers, and if
bound, then P contains
1. The axiom of choice for finite sets is equivalent to both
(ZLF1) and (ZLF2).
A key step in the proof depends
Bourbaki:2
upon the following lemma due to
Lemma (Bourbaki).
If P is a partially ordered set such that every
well-ordered subset of P has a least upper bound, and if <f>is a mapping
of P into itself such that x^<b(x) for every xEP, then <p(x) = x for some
xEP.
Proof
of Theorem
1. We shall verify the implications
-*(ZLF1)->(ZLF2)^(ACF).
(ACF)
Assume first that (ACF) holds and
consider a partially ordered set P with finitary covers. We suppose
that every well-ordered subset of P has a least upper bound but that
2 The lemma is an immediate consequence of Théorème 1 and Corollaires 1 and
2 of [l ] (cf. N. Bourbaki, Éléments de mathématique, Vol. I ; Théorie des ensembles,
Chapter III, Paris, 1956, pp. 48-49); no form of the axiom of choice is used in its
proof.
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224
R. L. BLAIR AND M. L. TOMBER
[April
no element of P is maximal. Consider first any element x£P;
by
(Fl) the set C(x) is not empty. We define the relation Rx on C(x) as
follows: For a, bEC(x),
aRxb in case for some positive integer m
there exist elements x¿£C(x) (i = l, • • ■ , m) such that
a ~
Xi •—' x2 f**>• • • ~
xm ~
b.
Then Rx is an equivalence
relation on C(x), and, for a£C(x),
denote by Pi [a] the equivalence class of C(x) with representative
It follows from (F2) that Pi[a]
is finite for each aEC(x).
SF= U {{Rx[a]:aE
C(x)):xE
we
a.
Then
P)
is a family of nonempty finite sets so that, by (ACF),
choice function / on 3\ For each xEP, set
there exists a
E(x) = {f(Rx[a]):aEC(x)).
It is clear that every pair of elements of P(x) have a common upper
bound in P, and hence, by (F3), P(x) has a least upper bound </>(x)EPBut then the mapping x—><j>(x)has the property
that x<<¡>(x) for
every x£P,
which is contrary
to the lemma. We conclude that P
must have a maximal element and that (ACF) implies (ZLF1).
Since the implication
(ZLF1)—>(ZLF2)
is trivial, let us assume
(ZLF2) and consider a family S of nonempty finite sets. Denote by
P the set of all / such that / is a choice function on some subfamily
of 5F. Partially order P by defining/^g
in case g is an extension of/,
and, îorfEP,
denote by SD(/) the domain of/. It is easy to see that
every totally ordered subset of P has a least upper bound. Moreover,
it is clear that a function/GP
is a choice function on S if (and only if)
/ is maximal in P. To verify (ACF) it will therefore suffice to show
that P satisfies conditions (F1)-(F3).
We consider
a nonmaximal
element/GP.
Then
SÇ£2D(/) f°r some
SES. Choosing an element xES, define g(S) =x and g(T) =f(T) for
PG£>(/). Then gEC(f) and condition (Fl) is satisfied.
Next, it is easy to see that gEC(f) if and only if S(g) = £)(/)U {SB}
for some (unique)
g, hEC(f),
set S0ES —©(/),
from which it follows that,
for
g^h if and only if Sg = Sh- Thus ~ is an equivalence rela-
tion on C(f). We claim that each '-«-'-equivalence class Q of C(f) is
finite. To see this, let gEQ and suppose that Sh^Sg for some hEQ-
If we define k(Sg) = g(Sg), k(Sh)=h(Sh), and k(T) =f(T) for PG £>(/),
then kEP
hEQ- But
now clear
Finally,
and g^k, h^k, a contradiction.
Hence Sh = Sg for every
S0 is finite, from which we conclude that Q is finite. It is
that P satisfies condition (F2).
let Xbea
nonempty subset of C(f) such that every pair
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I960]
225
THE AXIOM OF CHOICE FOR FINITE SETS
of elements of X have a common upper
every pair of distinct functions g, ÂGA.
bound. Then Sg^Sh for
We can therefore define
a function &GP as follows: k(T)=f(T) for TÇi£>(f), and, for each
gGA, k(Sg) =g(Sg). It is then clear that k is the least upper bound
of X and condition
2. Some
(F3) is satisfied.
equivalent
formulations
The proof is now complete.
of (AC).
Let
us denote
by
(ZLF1*) and (ZLF2*) the modifications of (ZLF1) and (ZLF2),
respectively,
obtained
by replacing,
in their statements,
"least upper
bound" by "upper bound."
The following observation,
originally devised for a slightly different
purpose, was communicated
to us by Herman Rubin: Let P be any
partially ordered set, let N be the set of all positive integers with its
usual order, and partially order the cartesian product Q = PXN
lexicographically.
Then (as one easily sees) the following statements
are equivalent:
(i) Every totally ordered subset K of P has an upper
bound xGA- (ii) Every totally ordered subset of Q has an upper
bound.
Now for each element (x, «)G(?, C((x, «)) consists precisely of the
single element (x, « + 1). Hence Q trivially satisfies conditions (Fl)—
(F3). Since Q obviously has no maximal element, both (ZLF1*) and
(ZLF2*) imply the existence in P of a totally ordered subset K having no upper bound xGA, and from this we infer (AC). We therefore
have the following result:
Theorem 2. The axiom of choice is equivalent to both (ZLF1*) awd
(ZLF2*).
It is a familiar fact that (AC) is equivalent
to the "maximal chain
theorem": If P is a partially ordered set, then every chain ( = totally
ordered set) in P is contained in a maximal chain. In view of Theorem
1, it is therefore natural to consider the following restricted form of
the maximal chain theorem:
(MCTF) If P is a partially ordered set with finitary covers, then every
chain in P is contained in a maximal chain.
Theorem
3. The axiom of choice is equivalent to (MCTF).
Proof.3 Since (AC) obviously
(MCTF) and consider an arbitrary
implies
partially
(MCTF),
let us assume
ordered set P and a chain
CQP. Denote by Q the family of all chains in P and partially
order
(P= eW{p}
(Fl),
by inclusion.
It is clear that
(P satisfies condition
3 This is an adaptation of an argument due to Scott [ó]. We are indebted to Professor Rubin for calling our attention to Scott's argument and to its applicability
in
the present context.
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226
R. L. BLAIR AND M. L. TOMBER
and, since (P is a complete lattice, (P also satisfies (F2) and (F3).
Therefore (P contains a maximal chain X. such that CG3C, and hence
A = U {A : ACzXi~\e}
is a maximal chain in P containing
C. We
conclude that (MCTF) implies (AC).
References
1. N. Bourbaki, Sur le théorèmede Zorn, Arch. Math. vol. 2 (1949-1950) pp. 434437.
2. L. A. Henkin,
Metamathematical
theorems equivalent to the prime ideal theorems
for Boolean algebras, Bull. Amer. Math. Soc. Abstract 60-4-553.
3. J. Los and C. Ryll-Nardzewski,
Effectiveness
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theory for
Boolean algebras, Fund. Math. vol. 41 (1954) pp. 49-56.
4. A. Mostowski,
Über die Unabhängigkeil
des Wohlordnungssatzes
vom Ordnungs-
prinzip, Fund. Math. vol. 32 (1939) pp. 201-252.
5. -,
Axiom of choicefor finite sets, Fund. Math. vol. 33 (1945) pp. 137-168.
6. D. Scott,
The theorem on maximal
ideals in lattices and the axiom of choice,
Bull. Amer. Math. Soc. Abstract 60-1-172.
7. W. Sierpinski, L'axiome du choix pour les ensembles finis, Matematiche,
Catania
vol. 10 (1955) pp. 92-99.
8. Wanda
Szmielew,
On choices from finite sets, Fund.
Math.
vol. 34 (1947) pp.
75-80.
9. A. Tarski, Sur les ensemblesfinis, Fund. Math. vol. 6 (1924) pp. 45-95.
10. -,
Prime
ideal theorems for Boolean
algebras
Bull. Amer. Math. Soc. Abstract 60-4-562.
University of Oregon and
Michigan State University
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and the axiom
of choice,